Many instruments create an electronic signal that contains a frequency sweep or “chirp” that must be characterized. The frequency-swept optical signal generated by an absolute gravity meter provides a good example. Turning now to FIG. 1, the optical layout for a conventional absolute gravity meter in which one arm of a Michelson interferometer is in free-fall and the other arm of the interferometer is stationary, is shown. The light output L of a frequency-stabilized laser 10 is split into two beams B1 and B2 by a partially reflecting mirror or “beam splitter” 12. One of the beams B1 is reflected by the beam splitter 12 onto a mirror 14 in freefall. The other beam B2 passes through the beam splitter 12 and is reflected back towards the beam splitter 12 by a stationary mirror 16. The beam B1 reflected by the mirror 14 passes through the beam splitter 12 and is recombined with the beam B2 reflected by the mirror 16 to yield a return beam R. The return beam R impinges on a photodiode detector 18 resulting in an interference signal being generated. The interference signal exhibits a complete cycle of constructive and destructive interference whenever the paths of the two beams B1 and B2 change by one-half of the laser wavelength (λ/2). This change in intensity of the interference signal is referred to as a “fringe”.
As the mirror 14 accelerates, optical fringes are formed at the photodiode detector 18. The frequency of the interference signal depends upon the velocity of the mirror 14 according to Equation (1) below:f=2v/λ=2gt/λ  (1)
The mathematical form of the interference signal is a swept sinusoid for a Michelson-type interferometers Gravity causes the velocity of the falling mirror 14 to increase linearly with time. This in turn causes the frequency of the interference signal to increase as the mirror 14 falls. Initially, when the velocity of the mirror 14 is nearly zero, the fringe frequency is close to zero or DC but rises rapidly as the mirror 14 accelerates. As an example, using red laser light which has a frequency in the order of 633 nm and a typical drop distance of about 20 cm, the frequency will sweep (or “chirp”) from DC to approximately 6 MHz over the 0.2 s it takes the mirror 14 to fall. The interference signal V(t) is represented by Equation (2) below:
                              V          ⁡                      (            t            )                          =                  A          ⁢                                          ⁢                      sin            ⁡                          [                                                                    4                    ⁢                    π                                    λ                                ⁢                                  (                                                            x                      0                                        +                                                                  v                        0                                            ⁢                      t                                        +                                                                  1                        2                                            ⁢                                              gt                        2                                                                              )                                            ]                                                          (        2        )            where:
x0 and v0 are the initial position and velocity of the mirror; and
g is the gravitational acceleration of a freely falling body.
As will be appreciated by those of skill in the art, the interference signal is a challenging signal to measure because it is spread over a large bandwidth in the frequency domain but is only present for a short amount of time.
Traditionally, this interference or chirped signal has been analyzed using a method that measures time intervals between zero-crossings of the interference signal. FIG. 2 shows an interference signal generated by the absolute gravity meter of FIG. 1 and its zero-crossing times. Because each zero-crossing corresponds to the mirror 14 falling a distance equal to λ/2, the time and distance for each zero-crossing are related by Equation (3) for constant acceleration as expressed below:
                              x          n                =                              n            ⁢                                                  ⁢                          λ              2                                =                                    1              2                        ⁢                          gt              n              2                                                          (        3        )            
A linear least-squares analysis is used to fit for the unknown gravity value. Typically, an arbitrary initial mirror position and initial mirror velocity are included in the fit to allow for offsets in the mirror position and non-zero initial velocity. The model function also usually includes small corrections due to frequency modulation of the laser, gradient of gravity, and even a correction due to the finite speed of light as described in the publication entitled “A New Generation of Absolute Gravimeters”, authored by Niebauer et al., Metrologia Vol. 32, Num. 3, 1995.
One advantage of the above zero-crossing method is that the fit is linear in the parameters of initial position, velocity, and acceleration of the mirror 14. Another advantage is that the result is, in principle, independent of the amplitude of the interference signal. The zero-crossing method was feasible even before the advent of very fast computers as little data is required to make the fit and time interval analyzers that could make high speed time measurements were available in the early 1970s. For example, the JILAG series of absolute gravity meters (1980-1985) typically scaled the zero-crossings by about 4000 so that only about 100 points were fit during each mirror drop. The FGS absolute gravity meter (2005) typically uses about 600-700 data points per mirror drop. Higher data acquisition rates are possible, but the effect on noise reduction is minimal because of small systematic errors in the interference signal.
Although the zero-crossing method presents some advantages, there are several disadvantages associated with the zero-crossing method. In particular, the acquired data are equally spaced in mirror freefall distance but are not equally spaced in the time domain. This makes it difficult to study certain types of noise or look for vibration signals in the time domain. Another difficulty is that the zero-crossing detection system must avoid double triggering and therefore, requires a relatively clean input interference signal. This puts a lower limit on the amount of light needed in the interferometer for the instrument to function properly.
Another technique used to analyze au interference signal digitizes the interference signal and then fits the data for the amplitude, the initial phase, the initial frequency, and the frequency chirp of the interference signal. This technique was firstly employed in an absolute gravity meter in 1979 as described in the publication entitled “A Transportable Apparatus for Absolute Measurement of Gravity” authored by Murata, Bulletin of the Earth-quake Research Institute 53: 49-130, 1979. As described, very accurate photographs of an interference signal waveform on an oscilloscope were analyzed and digitized by hand. The procedure was obviously slow and tedious, but it did accomplish the task of determining gravity.
Digitizing the interference signal waveform has become more attractive as the technology of analog-to-digital (A/D) converters has advanced. Currently it is possible to digitize an interference signal waveform at frequencies close to 1 GHz, and high speed computers can now automate the non-linear fitting routines needed to determine gravity. However, it is important to note that a large dynamic range is also needed as typical gravity meters try to reach a resolution in the order of 10−9 or even 10−10 g in one mirror drop.
U.S. Pat. No. 5,637,797 to Zumberge et al. describes digitizing an interference signal waveform automatically using a fast A/D personal computer (PC) card, but sampling well below Nyquist (1 MHz). A standard non-linear least-squares fit to Equation (1) is performed. The mirror velocity term appears as an initial frequency, and the gravity term is the frequency chirp. Unfortunately, it is extremely difficult to provide initial estimates of the velocity and gravity accurately enough so that the non-linear fit will converge. This becomes increasingly difficult as the sampling frequency is decreased. Zumberge et al. determines apparent zero-crossings and uses first differences of mean-time for zero-crossings to deduce the initial velocity and acceleration (equivalent to initial frequency and chirp) in the time domain. Apparently, 85,000 points from a single drop in 5.3 s can be fit with a 50 MHz 80486 personal computer.
The publication entitled “A Method to Estimate the Time-Position Coordinates of a Free-Falling Test-Mass in Absolute Gravimetry” authored by D'Agostino et al., Metrologia 42 (2005) 233-238 discloses a similar method but where the interference signal waveform is digitized above Nyquist at 50 MHz. Unfortunately, both the Zumburge et al. and D'Agostino et al. methods require very large amounts of data and a large number of operations in order to determine accurately the parameters in the nonlinear least-square's fit. For example, if one (1) million points are sampled during each mirror drop, about 2 MB of data storage is needed. In a typical absolute gravity measurement, such a mirror drop is performed every ten (10) seconds, resulting in 1 GB of data being generated every three (3) hours. Even with modern computers, the algorithms proposed by Zumberge et al. and D'Agostino et al. can take many seconds to process one measurement. For example, as D'Agostino et al. state “ . . . the program processes about 750 windows of the interference signal in about 35 s on the adopted computer, which has 1 GB of RAM and uses a 1.8 GHz Pentium IV processor.”
As will be appreciated, improvements in measuring chirped signals such as an interference signal generated by an absolute gravity meter are desired. It is therefore an object of the present invention to provide a novel method and apparatus for processing an under-sampled chirped sinusoidal waveform using a complex-heterodyne.